Atoms are attracted to each other by weak dispersion forces (also called London forces), linked to small oscillating dipoles created by electron cloud fluctuation. Atoms are also repelled by each other due to the Pauli exclusion principle, which states that no two electrons may share the same quantum numbers. Fortuitously, dispersion forces act over a longer distance (becoming stronger linearly as 1/r to the sixth power increases), while repulsion forces only act over a short distance (becoming more repulsive linearly as 1/r to the twelfth power increases.) This observation is expressed in the Lennard-Jones potential (equation C.1), which calculates the energy of interaction as a combination of attractive and repulsive forces:
A and B are variables that are determined experimentally for each atom or functional group. The distance between two atoms or atom groups at which E = 0 defines the van der Waals radii for those two groups. Since atoms will choose to interact with one another at a distance that gives the most negative energy of interaction, Em, which varies between -0.01 kcal/mol and -0.25 kcal/mol. Rm (the distance that corresponds to the minimum energy of interaction) is often somewhat longer that the van der Waals (vdW) radius (see Figure C.1).
Figure C.1 Plot of the Lennard-Jones potential vs. distance. Note that Rm represents the distance that gives the minimum energy of interaction and that double the vdW radius describes the separation at zero energy of interaction.
Table C.1 includes data for van der Waals radii and one-half Rm (the value of 1/2 Rm provides a radius comparable to the van der Waals radius), reflecting minimum and optimal distances of separation, respectively. These values can be helpful in identifying favorable and unfavorable interactions within a large biological molecule and between such molecules.
Table C.1 Distances associated with van der Waals contacts between various atoms and atom groups.(1) Note that hydrogen atoms are included in the van der Waals radii for those groups that contain them.
| Group | vdW radius | 0.5 Rm |
| C atom | ||
| - aliphatic | 1.7 Å | 2.0 Å |
| - aromatic | 1.7 Å | |
| O atom | ||
| - carbonyl | 1.4 Å | 1.9 Å |
| - alcohol | 1.5 Å | |
| N atom | ||
| - amide | 1.52 Å | 1.80 Å |
| - amine | 1.65 Å | |
| - ammonium | 1.50 Å | |
| F atom | 1.35 Å | |
| Cl atom | 1.80 Å | |
| Br atom | 1.95 Å | |
| I atom | 2.15 Å | |
| S atom | 1.85 Å |
The ideal bond angle for a hydrogen bond is 180 degrees. The three atoms in the X-H...X interaction should lie on a line. In reality, though, the angle is typically has a value of 160 ± 20 degrees. Directionality is not key, since the hydrogen bond is relatively "soft" and does not show a great deal of energetic variation over a reasonable range of angles. In essence, the energy associated with an H-bond is not sufficient to dictate structure on its own (as a covalent bond would) but instead reflects some constraints associated with its environment.
Bond distances between heteroatoms involved in hydrogen bonding are equally variable. Ideally, most H-bonds would exist between atoms separated by 2.7 Å to 3.1 Å, but that distance depends on the atoms involved and the structural constraints imposed by the environment. Here are some typical bond distances:
Table C.2 Distances associated with hydrogen bonds between various atoms and atom groups.(2)
| Bond | Ave. Dist. (Å) | Range (Å) |
| N-H...N | 3.10 | 2.88-3.38 |
| N-H...O | ||
| - Amide NH | 2.93 | 2.55-3.04 |
| - Amino NH | 3.04 | 2.57-3.22 |
| N-H...F | 2.78 | 2.62-3.01 |
| N-H...Cl | 3.21 | 2.91-3.52 |
| O-H...N | 2.80 | 2.62-2.93 |
| O-H...O | ||
| - Alcohol OH | 2.74 | 2.55-2.96 |
| - Water OH | 2.80 | 2.65-2.93 |
| O-H...Cl | 3.07 | 2.86-3.21 |
(1) Data is taken from Schulz and Schirmer, Principles of Protein Structure, Springer-Verlag, 1979 and Lesk, Protein Architecture, IRL Press, 1991.
(2) Taken from Jeffrey and Saenger, Hydrogen Bonding in Biological Structures, Springer-Verlag, 1991, p. 29